Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

On Dwork’s $p$-adic formal congruences theorem and hypergeometric mirror maps

About this Title

E. Delaygue, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France, T. Rivoal, Institut Fourier, CNRS et Université Grenoble 1, CNRS UMR 5582, 100 rue des maths, BP 74, 38402 St Martin d’Hères cedex, France and J. Roques, Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des maths, BP 74, 38402 St Martin d’Hères cedex, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1163
ISBNs: 978-1-4704-2300-1 (print); 978-1-4704-3635-3 (online)
Published electronically: October 13, 2016
Keywords: Dwork’s theory, generalized hypergeometric functions, $p$-adic analysis, integrality of mirror maps
MSC: Primary 11S80; Secondary 14J32, 33C70

View full volume PDF

View other years and numbers:

Table of Contents


  • 1. Introduction
  • 2. Statements of the main results
  • 3. Structure of the paper
  • 4. Comments on the main results, comparison with previous results and open questions
  • 5. The $p$-adic valuation of Pochhammer symbols
  • 6. Proof of Theorem
  • 7. Formal congruences
  • 8. Proof of Theorem
  • 9. Proof of Theorem
  • 10. Proof of Theorem
  • 11. Proof of Theorem
  • 12. Proof of Theorem
  • 13. Proof of Corollary


Using Dwork’s theory, we prove a broad generalization of his famous $p$-adic formal congruences theorem. This enables us to prove certain $p$-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number $p$ and not only for almost all primes. Furthermore, using Christol’s functions, we provide an explicit formula for the “Eisenstein constant” of any hypergeometric series with rational parameters.

As an application of these results, we obtain an arithmetic statement “on average” of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.

References [Enhancements On Off] (What's this?)